Can compression algorithms be employed for recovering signals from their underdetermined set of linear measurements? Addressing this question is the first step towards applying compression algorithms to compressed sensing (CS). In this paper, we consider a family of compression algorithms $\mathcal{C}_R$, parametrized by rate $R$, for a compact class of signals $\mathcal{Q} \subset \mathds{R}^n$. The set of natural images and JPEG2000 at different rates are examples of $\mathcal{Q}$ and $\mathcal{C}_R$, respectively. We establish a connection between the rate-distortion performance of $\mathcal{C}_R$, and the number of linear measurement required for successful recovery in CS. We then propose compressible signal pursuit (CSP) algorithm and prove that, with high probability, it accurately and robustly recovers signals from an underdetermined set of linear measurements. We also explore the performance of CSP in the recovery of infinite dimensional signals. Exploring approximations or simplifications of CSP, which is computationally demanding, is left for the future research.